We have seen that the volume increases in proportion to the temperature and inversely in proportion to the pressure. Furthermore, it is clear that the volume of a gas must also be proportional to the number of particles present. Accordingly, all quantities can be combined as follows:
$$
V \propto \frac{N T}{p}
$$
Rearranging the equation and introducing the so-called Boltzmann constant \(k_\mathrm{B}\) with the value
$$
k_\mathrm{B} = 1.381 \cdot 10^{-23}\,\mathrm{J}{K}
$$
as a proportionality factor, we obtain an equation that correctly describes all state changes of the ideal gas:
$$
pV = N k_\mathrm{B} T
$$
This equation is often referred to as the ideal gas equation. For real gases, there are sometimes considerable deviations in different temperature and pressure ranges. Nevertheless, it will be used in the following as the starting point for further investigations.
The number of particles \(N\) in the gas equation can also be replaced by the amount of substance \(n\). In this case, the Boltzmann constant \(k_\mathrm{B}\) must be replaced by the gas constant \(R\):
$$
pV = nRT
$$
It can be easily shown by converting the particle number into the amount of substance that \(R\) must then take the value
$$
R = 8.31448\,\frac{\mathrm{J}}{\mathrm{K}\cdot\mathrm{mol}}
$$